Cancellation Laws in a Ring

Cancellation Laws in a Ring

We say that cancellation laws hold in a ring $$R$$ if
$$ab = bc\,\left( {a \ne 0} \right)\,\, \Rightarrow \,b = c$$ and $$ba = ca\,\,\left( {a \ne 0} \right)\,\,\, \Rightarrow b = c$$ where $$a,b,c$$ are in $$R$$.

Thus in a ring with zero divisors, it is impossible to define a cancellation law.



A ring has no divisor of zero if and only if the cancellation laws holds in $$R$$.


Suppose that $$R$$ has no zero divisors. Let $$a,b,c$$ be any three elements of $$R$$ such that $$a \ne 0,\,\,ab = ac$$.

\[\begin{gathered} ab = ac\, \Rightarrow ab – ac = 0 \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow a\left( {b – c} \right) = 0 \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow b – c = 0\,\,\,\,\,\,\,\,\,\,\,\,\left[ {{\text{because}}\,R\,{\text{is}}\,{\text{without}}\,{\text{zero}}\,{\text{divisor}}\,{\text{and}}\,a \ne 0} \right] \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow b = c \\ \end{gathered} \]

Thus the left cancellation law holds in $$R$$. Similarly, it can be shown that the right cancellation law also holds in $$R$$.

Conversely, suppose that the cancellation law holds in $$R$$. Let $$a,b \in R$$ and if possible let $$ab = 0$$ with $$a \ne 0,\,b \ne 0$$ then $$ab = a \cdot 0$$ (because $$a \cdot 0 = 0$$).

Since $$a \ne 0,\,\,ab = a \cdot 0 \Rightarrow b = 0$$

Hence we get a contradiction to our assumption that $$b \ne 0$$ and therefore the theorem is established.


Division Ring

A ring is called a division ring if its non-zero elements form a group under the operation of multiplication.

Pseudo Ring

A non-empty set $$R$$ with binary operations $$ + $$ and $$ \times $$ satisfying all the postulates of a ring except right and left distribution laws is called pseudo ring if
\[\left( {a + b} \right) \cdot \left( {c + d} \right) = a \cdot c + a \cdot d + b \cdot c + b \cdot d\] for all $$a,b,c,d \in R$$